Question: Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{t^2 + 9t}{t^2 + 2t - 63}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 + 9t}{t^2 + 2t - 63} = \dfrac{(t)(t + 9)}{(t - 7)(t + 9)} $ Notice that the term $(t + 9)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t + 9)$ gives: $z = \dfrac{t}{t - 7}$ Since we divided by $(t + 9)$, $t \neq -9$. $z = \dfrac{t}{t - 7}; \space t \neq -9$